No. You never want poor insulation. Poor insulation means that the indoor temperature will always match outdoor temperature. There's not such thing as "natural cooling", only thermodynamics.

Natural cooling is about sweat (water evaporation), and staying in the shade. Lots of wind helps.

It depends on the insulation type. If you're using mass as insulation, it can keep the house very cool. This is a common effect in stone houses and barns as they stay closer to the average daily/weekly tempature because it takes a long time for the tempature of that much mass to change.

Mass ≠ insulation. You can have mass as a buffer and regulator, but normally stone or brick and mortar houses simply shift the issues to later in the hot season. It is usually a more even temperature over any given timespan though, as you say.

I'd think barns would have very low mass in comparison to their size, but would stay close to surrounding temp by way of having very little insulation and being very "drafty"

Yes, they aren't the same but can be applied to a similar end goal.

It doesn't completely shift the issue to later in the season. Yes, it will shift some of it, but you still have the night/day average effect. This can still be beneficial, especially if in a shady location.

Barns can be drafty if you are talking about wooden barns. Stone barns, or stone foundation barns built into a hillside, can actually be cooler inside than outside during the heat of the day. That's part of why horse owner sometimes leave their animals in the barn during the day and turn them out at night (typical summer schedule).